Marek Biskup (University of California, Los Angeles)
Gibbs distributions on permutations over integers
Abstract:
I will discuss a problem that I learned from Daniel Ueltschi some 3 years
ago. Consider a probability distribution on the set of all permutations of
the integers that weighs a permutation by the exponential of the negative
sum of the squares of the displacements between the integers and their
images under the permutation. This problem arises as a caricature to Feynman's
representation of interacting Bose gases. I will show how to formalize the
above description in terms of infinite-volume Gibbs measures and then provide a full
classification of all such measures by means of the quantity called a flux.
In particular, all Gibbs measures are translation invariant and there is exactly
one that has only finite cycles, almost surely. The talk is based on joint work
-- and a paper under preparation -- with Thomas Richthammer (UCLA).